Laplace and Fourier Transforms
Teacher
Dr.Balaji Padhy
Category
Core Courses
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Course Name : Laplace and Fourier Transforms
Code(Credit) : CUTM1002(2-0-1)
Course Objectives
• To describe the ideas of Fourier and Laplace Transforms and indicate their applications in the fields such as application of PDE, Digital Signal Processing, Image Processing, Theory of wave equations, Differential Equations and many others.
• To use Fourier series for solving boundary value problems appearing in scientific & engineering problems.
Learning Outcomes
- Upon successful completion of this course, students will be able to:•Solve differential equations with initial conditions using Laplace transform.• Evaluate the Fourier transform of a continuous function and be familiar with its basic properties.
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COs
Course Outcomes
Mapping with COs with POs
(High-H, Low-L, Medium-M)
CO1 Have deep knowledge of Laplace Transformation and its real life application. PO1, PO2, PO5 CO2 Solve initial value problem and boundary value problem using Laplace Transform. PO1, PO2, PO4, PO5 CO3 Understand Fourier transform and its properties and will be able to solve the examples based on it. PO1, PO4, PO5 CO4 Have deep understanding to handle various types of problems using different kind of integral Transforms. PO1, PO2, PO5
Course Syllabus
Module-I(T-3 h + Pj-2 h)
Laplace Transforms, Properties of Laplace transforms, Unit step function.
Project-1
Make a short draft of properties of Laplace transform from memory. Then compare your notes with the text and write a report of 2-3 pages on these operations and their significance in applications.
Module-II(T-2 h + Pj-2 h)
Second shifting theorem, Laplace transforms of Derivatives and Integrals.
Project-2
Find the Laplace transform of the following functions(sinat,cos at,sinh at,cosh at,e^at,t^n,etc).
Module-III(T-3 h + Pj-2 h)
Derivatives and Integrals of Transforms, Inverse Laplace transform.
Project-3
Application of Unit step function (RC- Circuit to a single square wave).
Module-IV(T-2 h + Pj-2 h)
Solution of differential equations by using Laplace transform.
Project-4
Find the solution of differential equation by using Laplace Transform.
Module-V(T-4 h + Pj-2 h)
Periodic function, Fourier series, Fourier series expansion of an arbitrary period, Half range expansions.
Project-5
Find the Fourier series expansion of a 2-pi periodic function.
Module-VI(T-3 h + Pj-2 h)
Complex form of Fourier series, Fourier Integrals, Different forms of Fourier Integral.
Project-6
Find the Fourier sine and cosine integral of the following functions.
Module-VII(T-3 h )
Fourier Transforms, Fourier sine and cosine Transforms.
Text Book:
1. E. Kreyszig , Advanced Engineering Mathematics, Johnwilley & Sons Inc-8th Edition.Chapters:5(5.1 to 5.4(without Dirac's delta function ) ),10(10.1,10.4 and 10.7-10.9(definitions only , no proofs))
2. Highjer Engineering Mathematics by B.V.Ramana, Tata McGraw-Hill Education India, Inc-8th Edition
Reference Text Book:
1) Advanced Engineering Mathematics by P.V.O’ Neil Publisher: Thomson
2) Mathematical Methods by Potter & Goldberg ; Publisher : PHI
Session Plan
Session 1
Session 2
Properties of Laplace Transform
Lecture Notes Link: http://courseware.cutm.ac.in/wp-content/uploads/2020/08/Properties of Laplace transform.docx
Video Lecture Link:https://www.youtube.com/watch?v=pdRV1ICQnZ8video-2
Session 3
Unit Step Function
Lecture Notes Link: http://courseware.cutm.ac.in/wp-content/uploads/2020/08/unit-step-function.pdflec-3 pdf
Video Lecture Link:https://www.youtube.com/watch?v=wvV8Hq-4kDQ-
Session 4 & 5
Project-1
Make a short draft of properties of Laplace transform from memory. Then compare your notes with the text and write a report of 2-3 pages on these operations and their significance in applications
Session 6
Second Shifting Properties
Lecture Note Link: http://courseware.cutm.ac.in/wp-content/uploads/2020/08/SECOND-SHIFTING-THEOREM.pdf
Video Lecture Link https://www.youtube.com/watch?v=pdRV1ICQnZ8
Session 7
Properties of Laplace transform (Transform of Derivative and Integrals)
Lecture Notes:(Transform of Derivative and Integrals).
Video Lecture Link: https://www.youtube.com/watch?v=EsSBiIJGoO4
Session 8 & 9
Project-2
Find the Laplace transform of the following functions.
Session 10
Derivatives and Integrals of Transforms
Lecture Notes Link:http://courseware.cutm.ac.in/wp-content/uploads/2020/08/TRANSFORM-OF-INTEGRALS.pdf
video Lecture Link: https://www.youtube.com/watch?v=7uHkfustHo4
Session 11
Session 12
Session 13 & 14
Project-3
Application of Unit step function (RC- Circuit to a single square wave).
Session 15
Solution of Differential Equation by using Laplace transform.
Lecture Note Link: http://courseware.cutm.ac.in/wp-content/uploads/2020/08/Solution-of-differential-equation-by-Laplace-Transform.pdf
video Lecture Link: https://www.youtube.com/watch?v=kbp9qWS-Bsk
Session 16
Solution of initial value problem by using Laplace transform.
Lecture Note Link: http://courseware.cutm.ac.in/wp-content/uploads/2020/08/Solution-of-differential-equation-by-Laplace-Transform.pdf
video Lecture Link: https://www.youtube.com/watch?v=kbp9qWS-Bsk
Session 17 & 18
Project-4
Find the solution of differential equation by using Laplace Transform.
Session 20
Fourier series Expansion of a periodic functions
Lecture Note Link: http://courseware.cutm.ac.in/wp-content/uploads/2020/08/Fourier-series-expansion-of-periodic-function.pdf
video Lecture Link: https://www.youtube.com/watch?v=r6sGWTCMz2k
Session 21
Fourier expansion of a functions of any Periods 2L
Lecture Note Link: http://courseware.cutm.ac.in/wp-content/uploads/2020/08/Fourier-series-expansion-of-period-2L.pdf
video Lecture Link: https://www.youtube.com/watch?v=Ud9Xtxsi2HI
Session 22
Session 23 & 24
Project-5
Find the Fourier series expansion of a 2-pi periodic function.
Session 26
Fourier Integrals transform of a function f(t)
Lecture Note Link:http://courseware.cutm.ac.in/wp-content/uploads/2020/08/Fourier-Integral.pdf
Video Lecture Link: https://www.youtube.com/watch?v=ZZfRuPpRX-o
Session 27
Different forms of Fourier Integral Theorems.
Lecture Note Link:http://courseware.cutm.ac.in/wp-content/uploads/2020/08/Fourier-Integral-1.pdf
video Lecture Link: https://www.youtube.com/watch?v=ZZfRuPpRX-o
Session 28 & 29
Project-6
Find the Fourier sine and cosine integral of the following functions.
Session 30
Session 31
Fourier transform and Infinite Fourier Transforms
Lecture Note Link:https://www.slideshare.net/GMFahadBMostafa/fourier-transform-by-fabad-b-mostafa
video Lecture Link: https://www.youtube.com/watch?v=ykNtIbtCR-8
Case Studies
Our Main Teachers
Assistant Professor ,School of Applied Sciences, Centurion University of Technology and Management, Paralakhemundi, Odisha, India. He has over 11 years of teaching experiences. He has published 12 research articles in National and International refereed journals . He is continue their P.hD in the field of Fuzzy Inventory Control Models.